Rodgers says it would not be 'logical' to move right now.
He said: “I'll probably get asked again until they appoint someone, but it's very simple - I'm very happy here.
“I made a choice nine months ago to come here for a purpose.
“That was very clear in that I wanted to help this club move forward out of where it was when I came in, and looking to lead it forward with a group I feel has huge potential. Why would I want to move from that?
“But it's all about happiness and I couldn't be happier here.
“And I think logic would tell you – and I know that sometimes in your world it's not always logical this gossip and speculation – but why would I want to leave Leicester City at this moment in time?
“People may look at me thinking: 'Right, because a club like Arsenal, with the great institution and size, and all of that'… but it's all about happiness
“My contract is until 2022 and the club haven't said they want to release me from that. Take from that what you want.
“I gave an honest answer, which was based around probably most managers having something in their contracts, and, by all accounts, that opened the door for something.
“There's a lot for us to grow and develop over these next few seasons. We're moving to a new training facility."